from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8664, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,19,0,5]))
pari: [g,chi] = znchar(Mod(37,8664))
Basic properties
| Modulus: | \(8664\) | |
| Conductor: | \(2888\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2888}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8664.ce
\(\chi_{8664}(37,\cdot)\) \(\chi_{8664}(493,\cdot)\) \(\chi_{8664}(949,\cdot)\) \(\chi_{8664}(1405,\cdot)\) \(\chi_{8664}(1861,\cdot)\) \(\chi_{8664}(2317,\cdot)\) \(\chi_{8664}(2773,\cdot)\) \(\chi_{8664}(3229,\cdot)\) \(\chi_{8664}(3685,\cdot)\) \(\chi_{8664}(4141,\cdot)\) \(\chi_{8664}(4597,\cdot)\) \(\chi_{8664}(5509,\cdot)\) \(\chi_{8664}(5965,\cdot)\) \(\chi_{8664}(6421,\cdot)\) \(\chi_{8664}(6877,\cdot)\) \(\chi_{8664}(7333,\cdot)\) \(\chi_{8664}(7789,\cdot)\) \(\chi_{8664}(8245,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.321901219811890081790219546628722051791865953039568238015939027374467326085267423464178688376545784307644366848.1 |
Values on generators
\((2167,4333,5777,8305)\) → \((1,-1,1,e\left(\frac{5}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8664 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) |
sage: chi.jacobi_sum(n)